Ŕ Periodica Polytechnica Civil Engineering
60(2), pp. 247–255, 2016 DOI: 10.3311/PPci.8192 Creative Commons Attribution
RESEARCH ARTICLE
A Numerical Method for Eigensolution of Near-Regular Structural and
Mechanical Systems
Iman Shojaei, Ali Kaveh, Hossein Rahami
Received 30-04-2015, revised 26-06-2015, accepted 30-11-2015
Abstract
In this paper a numerical method is developed to find the eigenvalues of the Laplacian matrix for near-regular graph models. Considering the similarity between the pattern of the Laplacian matrix of a graph and the stiffness matrix of a struc- ture, the method can be used for the eigensolution of struc- tural/mechanical systems. Previously, using graph product rules algorithms for the swift solution of the equation F = K∆ in structural/mechanical systems were developed. In this study, using a purposeful ordering along with partitioning the Lapla- cian/stiffness matrix, a decomposable pattern is achieved. The decomposed parts are solved using combined graph product rules and numerical solutions. While the eigensolution of regu- lar patterns has been traditionally considered as an approxima- tion for near-regular patterns, here the solution of the regular pattern is utilized as an appropriate initial starting point for the solution of near-regular pattern. Furthermore, solving the de- composed regular part using graph product rules reduces the computational complexity of the method.
Keywords
Regular structures·Near-regular structures·Eigensolution· Graph products·Laplacian matrix
Iman Shojaei
Department of Biomedical Engineering, University of Kentucky, Lexington, KY 40506, USA
Ali Kaveh
Centre of Excellence for Fundamental Studies in Structural Engineering Iran University of Science and Technology, Narmak, Tehran, 1684613114, Iran e-mail: alikaveh@iust.ac.ir
Hossein Rahami
Engineering Optimization Research Group, College of Engineering; School of Engineering Science, College of Engineering, University of Tehran, Tehran, Ke- shavarz Blvd., No. 37, Ghods St., Iran
1 Introduction
Although the advent of powerful computers have smoothened the way for the swift computations structural/mechanical sys- tems, the analysis of complicated systems are still laborious and time-consuming. Many algorithms for the solutions of large- scale structural/mechanical systems have been developed over the past three decades. These algorithms have mainly aimed to find efficient solutions for the structural governing equations (i.e. F = K∆) and/or eigensolution of a system (i.e. frequen- cies of free vibration). However, due to the variety of sys- tems and lack of general patterns, most algorithms are lim- ited to partial applications. The most successful advancements were achieved in the solution of symmetric and regular pat- terns wherein linear algebra, graph products, group theoretical method, U-transformation etc. were employed to divide com- plicated large problems into sub-systems and solve the smaller parts with less computational complexity and then combine the solutions (i.e. divide and conquer methods) [1–7].
There are various structural/mechanical systems with geome- tries close to those of regular structures, but not satisfying the required mathematical conditions to be considered as regular. A model is called regular if it can be considered as the product of two or three graphs [1]. A near regular model consists of a reg- ular submodel with limited number of members and/or nodes being added or removed. Recent efforts have been devoted to realize, classify and solve these near-regular patterns efficiently through the available solution of the regular part [8–12]. While several flexibility and stiffness methods, and finite difference and finite element formulations were developed to solve the fi- nal governing equation F = K∆, less success were achieved for the eigensolution of the near-regular systems [8–12]. Compared to the solution of F = K∆, eigenvalue problems are more sensi- tive to the algebraic manipulations. This often leads to matrices with general patterns that cannot usually be solved using specific efficient solvers.
In this paper a numerical algorithm is presented for obtaining the eigenvalues of near-regular structures. In this method utiliz- ing the decomposition of block matrices, a determinant equation is obtained. This equation includes separate submatrices corre-
sponding to the regular and irregular parts of the system. The equation is then solved with a numerical method. The swiftness of the solution corresponds to: 1) Appropriate initial starting point of the solution (i.e. the eigenvalues of the regular struc- ture) and 2) Efficient solution of the separated matrix (corre- sponding to the regular system), in each iteration, using graph product rules.
2 Eigenvalues for the regular graph products
Sufficient conditions for converting Hermitian matrices A1
and A2into upper triangular ones using one orthogonal matrix are [11, 13, 14]:
A1A2=A2A1or A21=A22 (1) Consider the matrix M to be the sum of two Kronecker prod- ucts:
M=A1⊗B1+A2⊗B2 (2) If one of the conditions in Eq. (1) holds, the eigenvalues of the block matrix M are obtained as follows:
λM=
n
[
i=1
eig (Mi); Mi=λi(A1) B1+λi(A2) B2 (3) In this equation, the dimension of matrices A1and A2is equal to n, and the dimension of matrices B1and B2is equal to m.
In structural engineering block diagonal matrices, especially tri-block diagonal matrices, are of great importance. Some regu- lar forms of these matrices were solved previously [13, 15]. The following two forms correspond to the stiffness matrices of reg- ular structures:
Fn(Am,Bm,Cm)=
Am Bm
Bm Cm Bm Bm ... ...
... Cm Bm
Bm Am
;
Gn(Am,Bm,Cm)=
Am Bm Bm
Bm Cm Bm
Bm ... ...
... Cm Bm
Bm Bm Am
(4)
The solved patterns consist of the forms Gn(Am,Bm,Am), Fn(Am,Bm,Am) and the form Fn(Am,Bm,Cm) with the condi- tion Am −Bm = Cm. While the forms Gn(Am,Bm,Cm) and the form Fn(Am,Bm,Cm) with the condition Am−Bm,Cmare not completely regular to be swiftly solved.
3 Decomposition of partitioned block matrices
This section is devoted to some techniques of partitioned block matrices. Manipulating a partitioned matrix is a basic and helpful approach in matrix analysis. The applied methods in partitioned matrices are similar to those of ordinary numerical matrices in some ways. Consider a 2×2 matrix as
A=
a b c d
,a,b,c,d∈C (5)
Using a basic row operation, the second row of the matrix multiplied by 2 can be added to the first row so that the determi- nant of the matrix does not change:
1 2 0 1
a b c d
=
a+2c b+2d
c d
(6)
This row or column operation can be generalized to parti- tioned block matrices as
1 Interchanging two block rows or columns,
2 Multiplying a block row or column from the left or right by a matrix of proper size,
3 Multiplying a block row or column by a matrix and then adding it to another row or column.
It should be noted that for maintaining the symmetry of the matrix, these operations are performed simultaneously on rows and columns and therefore the determinant does not change.
Now using the above three basic rules one can convert a two- block matrix into a two-block diagonal matrix so that the deter- minant does not change:
In 0
−M21M−111 Im
M11 M12 M21 M22
In −M−111M12 0 Im
=
=
M11 0
0 M22−M21M−111M12
(7)
where M11is an invertible matrix of dimension n and M22is a matrix of dimension m
Consequently we will have
M11 M12
M21 M22
=det M11det
M22−M21M−111M12
(8)
For finding the eigenvalues of the partitioned matrices, the matrix M11is changed to the matrix M11 − λIn and the matrix M22is changed to the matrix M22 −λIm,
M11−λIn M12 M21 M22−λIm
=det (M11−λIn)·
·det
M22−λIm−M21(M11−λIn)−1M12
(9)
Equating this determinant to zero, results in:
det (M11−λIn)=0 (10) and
det
M22−λIm−M21(M11−λIn)−1M12
=0 (11)
Considering the condition of the decomposition where M11in Eq. (8) or M11 − λIn in Eq. (9) is invertible, Eq. (10) does not hold. Consequently, for finding the eigenvalues of the two-block matrix, only Eq. (11) should be solved.
As a simple example consider the following matrix
M=
0.8147 0.6324 0.9575 0.9572 0.9058 0.0975 0.9649 0.4854 0.1270 0.2785 0.1576 0.8003 0.9134 0.5469 0.9706 0.1419
Using Eq. (9), we will have
det
0.1576 0.8003 0.9706 0.1419
−λI−
−
0.1270 0.2785 0.9134 0.5469
0.8147 0.6324 0.9058 0.0975
−
−λI)−1
0.9575 0.9572 0.9649 0.4854
=0
The roots of this equation will be the eigenvalues of Matrix M
λM={2.4021,−0.0346,−0.7158,−0.4400}
Solving Eq. (9) results in finding the characteristic equation of the two-block matrix In general, finding the eigenvalues of matrices using their characteristic equation is not considered as a time-saving approach because both forming and solving the characteristic equation are difficult.
4 Computational complexity of eigensolution of near- regular graphs via solution of characteristic equation Eigensolution of a matrix by solving the equation det (M −λI) = 0 includes forming the characteristic equation and then solving it by an iterative method. In addition to the difficulties inherent to the formation of characteristic equation, using an iterative method for calculating eigenvalues makes the method very complicated. However, when it comes to near-regular forms, the method changes to an efficient approach via the application of the features of the regular patterns.
Complexity theory has been well developed in numerical lin- ear algebra. However, for eigenvalue problems where iterative methods are required, the complexity of algorithms cannot be easily obtained since the number of steps for achieving a de- sirable accuracy is not definite. Another problem is concerned
with the initial starting point that is very important in many it- erative algorithms which affects the number of iterations and consequently the computational complexity.
In this study, the difficulties of solving the equation det (M− λI) = 0, shown in Eq. (9) are smoothened for near- regular graphs. The method is carried out through
1 Removing the difficulty of inverting the matrix M11 − λIn, shown in Eq. (9).
2 Decreasing the complexity of the applied iterative method.
As it was mentioned, the matrices in this study are near- regular matrices composed of regular and irregular parts. Num- bering the nodes of the graph (structure) is performed in such a way that the regular and irregular parts are separated. There- fore the Laplacian (stiffness) matrix will have a two-block ma- trix form M as shown in Eq. (6), where the block M11represents the regular part and the block M22 represents the irregular part.
Such an ordering can be seen in Fig. 1 for a near-regular graph
Fig. 1.A near-regular graph with a suitable ordering
Where the black numbers form the regular block M11and the red ones form the irregular block M22. Previously, the inverse of the Laplacian matrix of regular graphs was obtained using its eigenpairs [15]:
Ax=b⇒ {ϕ}TjA{ϕ}jyj=λjyj={ϕ}Tjb yj= bj
λj
⇒ {x}n=
n
X
i=1
{ϕ}iyi=
n
X
i=1
{ϕ}ibi λi
=
n
X
i=1
{ϕ}i{ϕ}Ti λi
b (12) Whereλiand {ϕ}iare the eigenpairs of the matrix A.
Since the block M11 is regular, M11 − λIn is also regular.
But in references [13–15], the eigenpairs and the inverted ma- trix were found for a regular matrix filled with numbers. Invert- ing the parametrical matrix M11 −λIn, solving the parametrical determinant and finding the characteristic equation is rather dif- ficult. However, since the characteristic equation should be ul- timately solved iteratively to find the eigenvalues, in the present study an approach is utilized in which the iterative method is applied at earlier stage. In other words, the iterative solution is directly applied to Eq. (9) without forming the characteristic equation. The advantage of applying the iterative solution in ad- vance is that after making an initial guess forλin Eq. (9), the numerical matrix M11 − λInis quickly inverted using Eq. (10) and subsequently the determinant is solved. The determinant is
of the dimension of the irregular part (M22). Since in a near- regular graph we assume the irregular part to be small, the de- terminant is easily solved iteratively, until stopping criterion is satisfied. The iteration is performed using a bisection method to obtain all eigenvalues. Using strong initial guesses for theλis ensures the convergence of the solution. The Laplacian matrix for the graph shown in Fig. 1 can be written as
M=
M11 M12
M21 M22
; (M11)nm=Fn(Cm,Bm,Cm)= In⊗Cm+Tn⊗Bm
(13)
From Eq. (11) it can be seen that Tn =Fn(0,1,0), and obvi- ously InTn = TnIn,i.e. the condition of Eq. (1) is satisfied and M has the property of being block diagonalized.
According to the definition of a near-regular structure, the regular part is much larger than the irregular part. In other words, in the computational complexity of a near-regular struc- ture it is assumed that only the dimension of M11 (the regular part) approaches to infinity. Consequently, the initial guess is made using a matrix of the dimension of the matrix M but a pat- tern similar to the pattern of the regular matrix M11.Therefore, the following matrix is used for the initial guess
(M∗)lm =Fl(Cm,Bm,Cm) (14) Since the matrix M∗ has a regular form, its eigenvalues are easily obtained. The eigenvalues of the matrix M∗are close to the eigenvalues of the matrix M. The similarity becomes more obvious, when the regular part becomes larger.
Another way of finding the matrix M∗is to obtain the matrix M through a usual ordering as shown in Fig. 2. We will have
Mnm=Fn(Am,Bm,Cm) (15)
Fig. 2. A near-regular graph with a usual ordering
Since the graph follows the pattern of the strong Cartesian product, Am − Bm , Cmand it is classified as a near-regular form. For converting the near-regular matrix M to a regular one, the block Am is changed to the block Cm. This conver- sion causes the regular matrix M∗to be obtained again. In Ref.
[16] it is shown that addition of members around a graph, the condition Am − Bm = Cmholds and the eigenvaluses of the expanded graph are not much different with those of the graph
before addition. Thus the eigenvalues of M and M∗are close to each other.
Up to now, a proper ordering was considered, the iterative method was applied in advance, the appropriate initial guess was made and the inverse of the matrix M11 −λInwas found. Now, the complexity of the method is evaluated to show it is both measureable and less complicated. As it was mentioned, the computational complexity of many iterative methods is not mea- sureable. The computational complexity of an iterative bisec- tion method for finding the roots of the relationship f (x) = 0 is measureable only in the case of finding a certain root in the inter- val [a,b]. However, while finding the multiple roots of an equa- tion without having a sense of the domain of the roots, obtaining the complexity of the method is impossible. The problem gets worse when the order of the equation (the characteristic equa- tion in our problems) grows. Growing the order of the equation is certain because in calculating the complexity of the method the order of the equation (N) should approaches to infinity.
Now, it is shown why calculating the complexity of the method in the near-regular structures is possible. In a bisec- tion method for finding roots in the interval [a,b], the number of iterations (n) can be obtained as follows:
b−a
2n+1 < ε (16)
where ε is the upper bound for the error of the answer.
Eq. (14) can also be written as
n>ln (b−a)−ln(ε)
ln(2) −1 (17)
Which means for a specific ε, the number of iterations de- pends on the interval [a,b]. Consequently, the computational complexity of the method depends on the interval [a,b]. In ob- taining the computational complexity of an arbitrary character- istic equation finding the proper a and b is impossible because one cannot predict the changes of the interval [a,b] when the order of the equation grows. In many cases, by increasing the order of the equation, the interval [a,b] grows too with an indef- inite rate. This way, not only the number of iterations grows, but also finding the rate of the growth becomes impossible.
However, when we consider near-regular graphs, the interval [a,b] is limited and measurable. In a near-regular graph the reg- ular part is the dominant part regarding the definition of near- regular graphs. Moreover, the eigenvalues of the main struc- ture are close to those of the corresponding regular part. Con- sequently, for finding the domain of the problem (the interval [a,b]), one should obtain the domain of the regular structure, showing the lower and upper bounds are independent of the or- der of the equation. In graph product theory, the regular forms are presented in Kronecker forms, (Eq. 2).
Since the matrices A1 and A2 represent the number of sec- tors in a graph, in comparison with the unchanged matrices B1
and B2which represent the dimension of each sector, the matri-
ces A1 and A2 are the basic matrices and their dimension will increase when the graph M expands.
Therefore, the matrices A1and A2control the bounds of the eigenvalues. Consider the Fig. 3
Fig. 3. A near-regular structure with a suitable ordering
In such a near-regular structure the black nodes form the reg- ular matrix M11as
M11=F (C,B,C)=I⊗C+T⊗B;
T=F(0,1,0) (18)
The dimension of the matrices I and T represent the number of regular sectors that is equal to 4 in this example and the matri- ces B and C represent the number of nodes in each sector that is equal to 3 here. When the graph shown in the Fig. 3 grows, the number of sectors grows (I and T), while the number of nodes in each sector does not change (B and C). Thus the basic matrices A1and A2control the bounds and computational complexity.
The known basic matrices in graph products include two groups: 1) tri-diagonal matrices 2) Circulant matrices
Consider the following tridiagonal matrix for the first group:
Zn=
−α+b c 0 0 . . . 0 0
a b c 0 . . . 0 0
0 a b c . . . 0 0
. . . .
0 0 0 0 . . . a −β+b
n×n
(19)
The eigenvalues of such a matrix is presented in [17] using the following equation:
λ=b+2√
ac cosθθ,mπ,mZ (20) Whereθis calculated as follows:
ac sin (n+1)θ+(α+β)√
ac sin nθ+αβsin (n−1)θ=0 (21) In a regular graphαandβare equal to zero and a = c, then Eq. (19) is converted to
a2sin (n+1)θ=0 (22) And Eq. (18) changes to the following:
λ=b+2acosθθ,mπ,mZ (23)
Consequently, in a regular graph when the number of sectors increases, according to Eq. (21) the lower and upper bounds of the eigenvalues does not change. This is because n in Eq. (20) influence only the cosθin Eq. (21) which is always in the interval [-1, 1]. Therefore the lower and upper bounds of the eigenvalues in Eq. (21) are found. Using Eq. (3), the eigenvalues for a block tri-diagonal matrix is obtained.
Now, we consider the second basic group so-called circulant matrices:
P=
0 1 0 . . . 0 0 0 1 . . . 0 ... ... ... ... ... 0 0 0 . . . 1 1 0 0 . . . 0
n×n
(24)
The eigenvaluse of such a matrix are obtained via solving det (P−λI)=λn−1=0 (25) The lower and upper bounds of theλin Eq. (23) are simply obtained in the interval [-1, 1] for real values. These bounds are independent of the magnitude of the n. Again, using Eq. (3) the eigenvalues of a three-block diagonal matrix are found.
Now, according to Eq. (15), since the value of ln(b − a) is measurable and limited, the number of iterations are deter- mined for a regular graph with any arbitrary dimension. There- fore, the computational complexity of the iteration method is related to the number of roots that is n roots in a graph of di- mension n and consequently the complexity of the iteration step is O(n). It was mentioned that regular graphs hold the relation- ship A1A2 = A2A1. These matrices are tri-block diagonal ma- trices holding the patterns of Gn(AmBmCm) and Fn(AmBmCm) that decompose to n blocks of dimension m. Therefore, only n times the eigenvalues of matrices of dimension m should be ob- tained. Then, the inverse of the matrix M11 − λIn is obtained using the eigenvalues. It was mentioned as the dimension of the matrix M11 −λInapproach to infinity, m (the number of nodes in a sector) does not change and only n as the number of sec- tors grows. As a result, the complexity of inverting the matrix M11 −λInis also O(n). Again, consider the main equation as
det
M22−λIm−M21(M11−λIn)−1M12
=0 (26)
After obtaining the value of (M11 −λIn)−1for a specificλin an iteration the complexity of the term M21(M11 −λIn)−1M12
should be obtained. This is multiplication of three matrices in which the computational complexity is O(mn2) Since m is lim- ited, the computational complexity is O(n2) which is the largest and dominant complexity in solving this equation. Thus, an ef- ficient complexity is gained using the present method for the near-regular graphs. It is less complicated than other efficient known methods such as the method which uses a combination of Householder and QR methods with the complexity of O(n3)
Now, the present method is outlined in the following steps:
1 Ordering the near-regular graph so that first the regular part is numbered and then the irregular part.
2 Forming the Laplacian (stiffness) matrix for the graph (struc- ture). The matrix will have the pattern of the two-block matrix M because of the mentioned ordering.
3 Forming Eq. (9) as the main equation.
4 Forming a Laplacian (stiffness) matrix named M∗of the di- mension n + m (similar to the dimension of matrix M) but with the pattern of matrix M11.In other words, forming a reg- ular matrix corresponding to the near-regular matrix with the same dimension (see Eq. (12)).
5 Finding the eigenvalues of the regular matrix M∗ through a quick solution using Eq. (3). These eigenvalues are used as the suitable initial guesses as they are close to the eigenvaluse of the near-regular graph. The eigenvalues are in an interval [a,b] which is independent of the dimension of the matrix M∗.
6 Solving Eq. (9) through an iterative bisection method without forming the characteristic equation. The eigenvalues found in step 5 are used as initial guesses in this step. Applying the iterative method in advance makes the opportunity of solving (M11 −λIn)−1 quickly using Eq. (10), decreasing the com- plexity of the method.
Previously, the inverse of the stiffness matrix of near-regular structures was found. The inverted matrix can be calculated for a near-regular structure convertible to a regular form. These kinds of near-regular structures were presented in the references [8, 9].
In Eq. (9), the ordering is performed so that the matrix M11 − λInis regular and the condition A1A2 = A2A1 holds. There- fore, inverting this matrix using the eigenpairs and Eq. (10) is possible. However, since the stiffness matrix of near-regular structures can be efficiently inverted [8, 9], ordering of the ma- trix can also be performed so that the block M11is a near-regular matrix. Thus, the matrix M11 −λInis swiftly inverted while the condition A1A2 = A2A1does not hold. This way the present solution gets more generalized where the matrix M11 can be both regular and near-regular. The efficiency of the method is shown using some examples.
5 A Simple Illustrative Example
For the truss shown in the Fig. 4, the eigenvalues of the stiff- ness matrix are calculated. For all members the elastic modu- lus is considered as 210 kN/mm2and the cross section areas are taken as 15 cm2.
Based on the present algorithm, first a proper ordering is per- formed. According to Fig. 5, the truss is composed of regular and irregular parts that are shown in green and red colors, re- spectively.
Ordering of the regular part is performed sector by sector as it is shown in Fig. 5. In this structure each five nodes form a sec- tor, for example the first sector is composed of the nodes 1, 2,
Fig. 4. The truss under study
Fig. 5. The truss composed of regular and irregular parts with a suitable or- dering
3, 4 and 5. Thus, the matrix M11contains 7 blocks (each sector represents a block) of dimension 10 (each sector consists of 5 nodes with each node having two degrees of freedom). The last regular sector does not belong to the matrix M11 since it has 2 nodes in common with the irregular part (the nodes 38 and 40) that causes this sector not to follow the pattern of other regu- lar sectors. Consequently, the sector number 8 is considered as a part of the matrix M22 as well as the irregular part shown in the red color. The matrix M22contains 7 nodes and is of dimen- sion 14. Using this partitioning, the reduced stiffness matrix will have the following form:
M=
M11 M12 M21 M22
84×84
where M12,M21and M22are matrices of dimensions 70×14, 14×70 and 14×14, respectively, and
M11=F70×70(C,B,C)=I⊗C+T⊗B where B and C are matrices of the dimension 10.
For finding the initial guesses and the interval [a,b] the ma- trix M∗ should be formed. It was mentioned that this matrix should have dimension identical to that of the matrix M. But, since the dimension of the matrix M is 84, the matrix M∗cannot be formed because it is composed of blocks of dimension 10. In other words, this matrix just can have dimensions of the multi- ples of 10 such as 80 or 90. To overcome the problem, 6 rows and columns filled with zeros are added to the matrix M to get the dimension 90. These zeros do not affect the eigenvalues of the matrix M and only add 6 additional zero eigenvalues to the final answer. This new matrix is named M0
M0 =
M ... 084×6
· · · · 06×84 ... 06×6
90×90
The matrix M∗of the dimension 90 (9 blocks of the dimen- sion 10) is formed as:
M∗=F (C,B,C)=I⊗C+T⊗B
In fact the matrix M∗ is the stiffness matrix of the regular structure of Fig. 6.
Fig. 6. A regular structure used for appropriate initial guesses
The supports in the right side are chosen so that the last sector is similar to the other sectors.
Now using Eq. (3), the eigenvalues of the matrix M∗are ob- tained. There are 90 eigevalues for the matrix M∗. Typical eigenvaluesλ11toλ20are written in the following:
λM∗=106[1.7329, 1.8604, 2.0003, 2.0649, 2.4029, 2.4228, 2.9563, 3.4094, 3.5748, 3.9336]
Since for the matrix M∗, λmin = λ1=0.2951×106 and λmax = λ90=63.461×106and regarding the six zero eigenval- ues added to the main structure, the interval used in the bisection method is chosen as [0 65×106].
The initial guesses and the interval are ready to be utilized in the bisection method. All terms in the equation
det
M22−λIm−M21(M11−λIn)−1M12
=0
were calculated. Now, the equation is solved in each step using the bisection method. The eigenvaluesλ11toλ20are found as
λM=106[1.5934, 1.8151, 1.9138, 2.1311, 2.2158, 2.7474, 3.1877, 3.3628, 3.5226, 3.9020]
It can be observed thatλ∗Mis an efficient initial guess forλM. In order to obtain more precise answers, one can use Rayleigh-Ritz relationship by means of which having the ap- proximate eigenvalues and eigenvectors, one can get closer to the eigenvalues and eigenvectors of the main problem.
6 Numerical Examples
In this section different examples are studied to show the effi- ciency of the present method.
Example : In this example the frequencies of the free vibra- tion of the double layer grid, shown in the Fig. 7 are calculated.
A real case of such a grid can be seen in Fig. 8. For all mem- bers the elastic modulus is taken as 200 kN/mm2 and the cross section areas are 10 cm2. The mass is equal to 7.8 kg for the ver- tical and horizontal members and 11.0 kg for the diagonal ones.
The plan of the structure and its lateral view can be seen in the Figs. 9 and 10 respectively.
Fig. 7.A double layer grid with near regular pattern
Fig. 8.A real double layer grid
Fig. 9.The plan of the grid
To find the frequencies of the structure, the following equa- tion should be solved:
K−ω2iM
{∅}i={0}
Multiplying the two sides by M−1, we will have (M−1K−ω2iI){∅}i={0}
And then
det (M−1K−ω2iI)=0
Fig. 10. Lateral view of the grid
The structure is geometrically regular. However, it is consid- ered as a near-regular structure mathematically using a suitable nodal ordering. In this paper, a graph or structure is considered as regular if the pattern of its Laplacian matrix or stiffness ma- trix follows graph product rules. A graph or structure with just a regular geometry is not considered as regular. The structure is composed of a central regular part shown in green and an ir- regular part shown in red, Fig. 11. Using a proper ordering and considering M−1K = S, the matrix S of the dimension 192 with the following pattern is obtained:
S=
S11 S12
S21 S22
where S11is the block of the regular green part, and S22 cor- responds to the block of the irregular red part.
Using the blocks of the matrix , the main equation is formed det
S22−ω2Im−S21
S11−ω2In−1
S12 =0
The solution of this equation using the present algorithm re- sults in finding the values ofω. The first 10 frequencies are obtained as
ω=104[0.8791, 1.1253, 1.3324, 1.6811, 1.9371, 2.0916, 2.2983, 2.4577, 2.7348, 3.0060]
Fig. 11. The regular and irregular parts of a double layer grid.
Example : An important practical application of the present method is in the analysis of structures with constructional im- perfection. Consider the structure shown in Fig. 12. Because of constructional imperfection the nodes 31 and 35 are not in the right place as shown in Fig. 13. The amount of their deviation is seen in Fig. 13. For all members, the elastic modulus is taken
as 210 kN/mm2and the cross section areas are 15 cm2. The spe- cific gravity of steel is equal to 7800 kg/m3. The frequencies of the real structure are found.
The frequencies of the regular ideal structure shown in Fig. 12 are swiftly found using graph products. However, due to the constructional imperfection, the real structure changes into a near-regular one composed of regular and irregular parts as shown in Fig. 14.
Fig. 12. The ideal form of the truss under study
Fig. 13. The real form of the truss including constructional imperfection
Fig. 14. Constructional imperfection of the last story
To solve the real structure, the frequencies of the regular ideal structure can be used as an initial guess. The main equation is written as
det
S22−ω2Im−S21
S11−ω2In−1
S12 =0
Fig. 15. A structure composed of regular and irregular parts
where the matrix S11with dimension 50, represents the green regular part, and S22 with dimension 20, represents the red ir- regular part. Using Eq. (3), the frequencies of the ideal structure are calculated. The frequenciesω61toω70are obtained as
ω=104[15.3324, 15.7053, 16.1053, 16.4158, 16.7371, 17.0850, 17.3017, 17.5989, 17.7348, 17.9126]
Now, these frequencies are used as the initial guess in the bisection stage of the present algorithm. The frequenciesω61to ω70of the real structure are obtained as
ω=104[15.6824, 15.9358, 16.2489, 16.5298, 16.8626, 17.2007, 17.4611, 17.7218, 17.9346, 18.1043]
7 Concluding Remarks
In this paper an efficient algorithm is presented to calculate the eigenvalues (frequencies of free vibration) of near-regular structures composed of regular and irregular parts. The method is a combination of both exact and iterative solutions. The reg- ular and irregular parts get separated using a proper ordering and thus a two block matrix is obtained. This matrix is then de- composed using the rules of the partitioned block matrix and the main equation is obtained. The solution of the gained equation results in the eigenvalues. Instead of forming the characteris- tic equation and then applying the iterative method, the main equation is directly solved using an iterative bisection method.
This direct solution provides the opportunity for applying the graph product rules for a swift solution. The efficiency of the method depends on the computational complexity of the itera- tive method. For an arbitrary matrix where the graph product rules do not hold and there is no effective initial guess for the iterative method, the method is as efficient as the other existing approaches. However, when near-regular structures are consid- ered, since the graph product rules are set in the regular part and the initial guesses with limited bounds are available, the method
is less complex than other approaches. The presented method is applicable to calculating the vibrational frequencies of near- regular structures, rehabilitated structures, structures with con- structional imperfection and similar problems.
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